Crystal radii of ions

Crystal Radii and Univalent Crystal Radii of Ions... 

Since the electron distribution function for an ion extends indefi-finitely, it is evident that no single characteristic size can be assigned to it. Instead, the apparent ionic radius will depend upon the physical property under discussion and will differ for different properties. We are interested in ionic radii such that the sum of two radii (with certain corrections when necessary) is equal to the equilibrium distance between the corresponding ions in contact in a crystal. It will be shown later that the equilibrium interionic distance for two ions is determined not only by the nature of the electron distributions for the ions, as shown in Figure 13-1, but also by the structure of the crystal and the ratio of radii of cation and anion. We take as our standard crystals those with the sodium chloride arrangement, with the ratio of radii of cation and anion about 0.75 and with the amount of ionic character of the bonds about the same as in the alkali halogenides, and calculate crystal radii of ions such that the sum of two radii gives the equilibrium interionic distance in a standard crystal. 

Similar substantially constant differences are obtained with other pairs of alkali halides of B 1 structure, having either a cation or an anion in common. As a result, the conclusion was reached that each ion makes a specific contribution toward an experimentally observed r0, well-nigh irrespective of the nature of the other ion with which it is associated in the lattice. In other words, characteristic radii should be attributable to the ions (1,2). However, a knowledge of the internuclear distances in the crystals is not sufficient by itself to determine absolute values for crystal radii of ions, and various criteria have been used to assign the size of a particular ion or the relative sizes of a pair of alkali and halide ions. 

Since in all calculations the value n = 1 was chosen, the number of lattice sites per unit area M/A was calculated from the crystal radii of ions, assuming an hexagonal lattice structure. The bulk activity of the ions a] was approximated by the mean activity of the salt calculated from the Debye Huckel equation. Finally, an estimation of the magnitude of B/( a) shows that this parameter ranges from -3 to 0. Indeed if we assume that Ps = 6 10 °Cm, 1 = 5- 10 °m, M/A = 6 10 sites m and a = 2, then ... 

Since every atom extends to an unlimited distance, it is evident that no single characteristic size can be assigned to it. Instead, the apparent atomic radius will depend upon the physical property concerned, and will differ for different properties. In this paper we shall derive a set of ionic radii for use in crystals composed of ions which exert only a small deforming force on each other. The application of these radii in the interpretation of the observed crystal structures will be shown, and an at- Fig. 1.—The eigenfunction J mo, the electron den-tempt made to account for sity p = 100, and the electron distribution function the formation and stability D = for the lowest state of the hydr°sen of the various structures. 

In deriving theoretical values for inter-ionic distances in ionic crystals the sum of the univalent crystal radii for the two ions should be taken, and corrected by means of Equation 13, with z given a value dependent on the ratio of the Coulomb energy of the crystal to that of a univalent sodium chloride type crystal. Thus, for fluorite the sum of the univalent crystal radii of calcium ion and fluoride ion would be used, corrected by Equation 13 with z placed equal to y/2, for the Coulomb energy of the fluorite crystal (per ion) is just twice that of the univalent sodium chloride structure. This procedure leads to the result 1.34 A. (the experimental distance is 1.36 A.). However, usually it is permissible to use the sodium chloride crystal radius for each ion, that is, to put z = 2 for the calcium... 

Fig. 1.—The sizes of spherically-symmetrical ions. The radii of the spheres are taken equal to the crystal radii of the ions.

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

Zachariasen, W. H. (1931). A set of empirical crystal radii for ions with inert gas configuration. Z. Kristallogr. 80, 137-153. 

Crystallographic data (.continued) for transition metal tetrafluorides, 27 98 for transition metal trifluorides, 27 92 Crystallographic disorder, nitrosyl groups, 34 304-305 Crystallography fuscoredoxin, 47 380 prismane protein, 47 232-233 Rieske and Rieske-type proteins, 47 92-109 Crystal radii, of various ions, 2 7 Crystals, 39 402 Crystal structure actinide metals, 31 36 copper-cobalt supetoxide dismutases, 45 ... 

The crystal radii of multivalent ions, such that the sum of two crystal radii is equal, to the actual equilibrium interiomc distance in a crystal containing the ions, can be calculated from their univalent radii by multiplication by a factor obtained from consideration of Equation 13-3. From this equation it is seen that the equilibrium interionic distance in a crystal containing ions with valence, is... 

Anion Contact and Double Repulsion.2 —The explanations of the deviations from additivity are indicated by Figure 13-6, in which the circles have radii corresponding to the crystal radii of the ions and are drawn with the observed interionic distances. It is seen that for LiCl, LiBr, and Lil the anions are in mutual contact, as suggested in 1920 by Land6.14 A simple calculation shows that if the ratio p = r+/r of the radii of cation and anion falls below /2 — 1 = 0.414 anion-anion contact will occur rather than cation-anion contact (the ions being considered as rigid spheres). A comparison of apparent anion radii in these crystals and crystal radii from Table 13-8 is given in Table 13-7. 

Crystals with the Rutile and the.Fluorite Structures Interionic Distances for Substances of Unsymmetrical Valence Type.—In a crystal of a substance of unsymmetrical valence type, such as fluorite, CaFs (Fig. 13-10), the equilibrium cation-anion interionic distance cannot be expected necessarily to be given by the sum of the crystal radii of the bivalent calcium ion and the univalent fluoride ion. The sum of the univalent radii of calcium and fluoride, 2.54 A, would give the equilibrium interionic distance in a hypothetical crystal with attractive and repulsive forces corresponding to the sodium chloride arrangement. 

LANTHANIDE CONTRACTION. The decreasing sequence of crystal radii of the triposiiivc rare-earth ions with increasing atomic number in the group of elements 157) lanthanum through (71) lutetiunt of the Lanthanide Series in the periodic table. 

The crystallographic ionic radii of the rare-earth elements in oxidation states +2 (CN = 6), +3 (CN = 6), and +4 (CN = 6) are presented in Table 18.1.3. The data provide a set of conventional size parameters for the calculation of hydration energies. It should be noted that in most lanthanide(III) complexes the Ln3+ center is surrounded by eight or more ligands, and that in aqueous solution the primary coordination sphere has eight and nine aqua ligands for light and heavy Ln3+ ions, respectively. The crystal radii of Ln3+ ions with CN = 8 are listed in Table 18.1.1. 

It would appear that the radii for alkali and halide ions based on experimental electron distribution results for NaCl provide the most realistic set currently available. The values agree well with crystal radii of the ions determined by Fumi and Tosi (77, 26) using the Born model of ionic solids in conjunction with solid-state data for the NaCl-type alkali halides — Table 2. It would be of considerable value if experimental electron distribution data in additional alkali halides were available to confirm the figures, but it appears that there can be important difficulties involved in their measurement (79)3. 

Table 2. Average crystal radii of the alkali and halide ions in the NaCl-type alkali halides obtained using the Born model (Born-Mayer form) (26)...

Blandamer and Symons (57) assumed in their work that the free energies of hydration of Rb+(g) and Cl (g) were identical, since the two ions have crystal radii of the same magnitude. Jain (58) has also applied the radii to calculate absolute free energies of ionic hydration. His treatment is in category (ii) and is based on the model developed by Frank and coworkers (59). The equation used is similar to that employed by Stokes (60) and utilizes calculated van der Waals radii of the ions as well as the crystal radii. To obtain the best agreement it was necessary to assume that the effective dielectric constant in water is 2.7. 

Since this work was essentially completed Meisalo and Inkinen (61) have reported a detailed X-ray diffraction analysis of potassium bromide. The crystal radii of the constituent ions were evaluated as Rk+ =1.57 0.07 A and =1.73 0.07 A. These magnitudes are compatible with... 

The radii of the atoms of the inert gases (in the condensed state) are not directly comparable with the radii of ions (in crystals) because of the much lower cohesion energy in the former case. 

For example, potassium varied by a factor of almost 10. Various authors found millimeter-sized fragments of this foreign material which represents the second component in the Apollo 12 soil samples28. As the foreign component was rich in potassium, rare earth elements (REE) and phosphorus, the acronym KREEP was devised. Later many other elements were found to be enriched in KREEP, for example, U, Th, Hf, Zr, Nb, and Ta. All these elements have in common large crystal radii (large-ion lithophile elements, LIL elements). The rock type from which the KREEP fragments were derived of is norite. 

The crystal radii of the rare earth metal ions decrease in a regular manner along the series. There is vast data suggestive of the formation of predominantly ionic complexes in the case of rare earth ions. Based on electrostatic theory, a direct relationship between the stability constant values and the atomic number of the rare earth metal ion is predicted [12]. In most of the complexes, this correlation of log K with Z holds good for La to Eu although in some cases the europium complexes are less stable than the samarium complexes. Further, this simple relationship is not valid when the heavy rare earth ions Tb to Lu are considered. 

This equation was deduced in Section 4.4.8. It is of interest to inquire here about its degree of appiicabiiity to ionic liquids, i.e., fused salts. To make a test, the experimental values of the self-diffusion coefficient D and the viscosity tj are used in conjunction with the known crystal radii of the ions. The product D r//T has been tabulated in Table 5.22, and the plot of D tj/T versus 1/r is presented in Fig. 5.31, where the line of slope k/6n corresponds to exact agreement with the Stokes-Einstein relation. ... 

Bonding within the silicate layers is predominantly ionic. As a result, forces are undirected and ion size plays an important role in determining crystal structure. Table 5.1 shows the crystal radii of common ions in silicates. The distance between two adjacent ions in a crystal can be measured accurately by x-ray methods. From a series of such measurements between different ions, the effective contributing radius of each ion can be determined. An ion has no rigid boundary an ion s radius depends on the number of its orbital electrons and on their relative attraction to the ion s nucleus. The radius of Fe ions, for example, decreases from 0.074 to 0.064 ran... 

Approximate values of n are known for various types of ions and are used to calculate the crystal radii of Table III.7... 

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